Plate&Shell Elements

Implementation of constant curvature triangular element for plate analysis. Formulation based on Mindlin hypothesis. The structure should be defined in x,y plane. The nodes should be numbered anti-clockwise (positive rotation around z-axis). The element features are summarized in Table 14.


Keyword	cctplate
Description	2D constant curvature triangular plate element
Specific parameters	-

Unknowns	Three dofs (w-displacement, u and v - rotations) are required in each node.
Approximation	Linear approximation of ratations, quadratic approximation of displacement.
Integration	Integration of all terms using one point formula.
Features	Layered cross section support.
CS properties	Cross section thickness is required.
Loads	Body loads are supported. Boundary loads are not supported now.
Output	On output, the generalized strains are printed in a vector with 12 components, with the following meaning: $e = {εx,εy,εz,γyz,γzx,γxy,κx,κy,κz,κyz,κxz,κxy},$ where ε_x,ε_y,ε_z are membrane normal deformations, γ_yz,γ_zx,γ_xy are shear componets, and κ_x,κ_y,κ_z,κ_yz,κ_xz,κ_xy are curvatures. The generalized stress vector contains corresponding integral forces/moments: $s = {nx,ny,ny,vyz,vzx,vxy,mx,my, mz,myz,mxz, mxy}.$ Please note, for example, that bending moment m_x is defined as m_x = ∫ σ_xz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo	0.
Status	Reliable

Table 14: cctplate element summary

2.6.2 CCT3D Element

Implementation of constant curvature triangular element for plate analysis. Formulation based on Mindlin hypothesis. The element could be arbitrarily oriented in space. The nodes should be numbered anti-clockwise (positive rotation around element normal). The element features are summarized in Table 15.


Keyword	cctplate3d
Description	Constant curvature triangular plate element in arbitray position
Specific parameters	-

Unknowns	Six dofs (u,v,w-displacements and u,v,w rotations) are in general required in each node.
Approximation	Linear approximation of ratations, quadratic approximation of displacement.
Integration	Integration of all terms using one point formula.
Features	Layered cross section support.
CS properties	Cross section thickness is required.
Loads	Body loads are supported. Boundary loads are not supported now.
Output	On output, the generalized strains are printed in a vector with 12 components, with the following meaning: $e = {ε ,ε ,ε ,γ ,γ ,γ ,κ ,κ ,κ ,κ ,κ ,κ }, x y z yz zx xy x y z yz xz xy$ where ε_x,ε_y,ε_z are membrane normal deformations, γ_yz,γ_zx,γ_xy are shear componets, and κ_x,κ_y,κ_z,κ_yz,κ_xz,κ_xy are curvatures. The generalized stress vector contains corresponding integral forces/moments: $s = {nx,ny,ny,vyz,vzx,vxy,mx,my, mz,myz,mxz, mxy}.$ Please note, for example, that bending moment m_x is defined as m_x = ∫ σ_xz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo	0.
Status	Reliable

Table 15: cctplate3d element summary

2.6.3 RerShell Element

Combination of CCT plate element (Mindlin hypothesis) with triangular plane stress element for membrane behavior. The element curvature can be specified. Although element requires generally six DOFs per node, no stiffness to local rotation along z-axis (rotation around element normal) is supplied. The element features are summarized in Table 16.


Keyword	rershell
Description	Simple shell based on combination of CCT plate element (Mindlin hypothesis) with triangular plane stress element. element can be arbitrary positioned in space.
Specific parameters	-

Unknowns	Six dofs (u,v,w-displacements and u,v,w rotations) are in general required in each node. Note, that although element it requires generally six DOFs per node, no stiffness to local rotation along z-axis (rotation around element normal) is supplied.
Approximation	Linear approximation of ratations, quadratic approximation of displacement.
Integration	Integration of all terms using one point formula.
Features	Layered cross section support.
CS properties	Cross section thickness is required.
Loads	Body loads are supported. Boundary loads are not supported now.
Output	On output, the generalized strains are printed in a vector with 12 components, with the following meaning: $e = {εx,εy,εz,γyz,γzx,γxy,κx,κy,κz,κyz,κxz,κxy},$ where ε_x,ε_y,ε_z are membrane normal deformations, γ_yz,γ_zx,γ_xy are shear componets, and κ_x,κ_y,κ_z,κ_yz,κ_xz,κ_xy are curvatures. The generalized stress vector contains corresponding integral forces/moments: $s = {nx,ny,ny,vyz,vzx,vxy,mx,my, mz,myz,mxz, mxy}.$ Please note, for example, that bending moment m_x is defined as m_x = ∫ σ_xz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo	0.
Status	Reliable

Table 16: rershell element summary

2.6.4 tr_shell01 element

Combination of CCT3D plate element (Mindlin hypothesis) with triangular plane stress element for membrane behavior. It comes with complete set of 6 DOFs per node. The element features are summarized in Table 17.


Keyword	tr_shell01
Description	Triangular shell element combining CCT3D plate element (Mindlin hypothesis) with triangular plane stress element with rotational DOFs
Specific parameters	-

Unknowns	Six dofs (u,v,w-displacements and u,v,w rotations) are in general required in each node.
Approximation	See description of cct and trplanstrrot elements
Integration	Integration of all terms using one point formula.
Features	Layered cross section support.
CS properties	Cross section thickness is required.
Loads	Body loads are supported. Boundary loads are not supported now.
Output	On output, the generalized strains are printed in a vector with 12 components, with the following meaning: $e = {εx,εy,εz,γyz,γzx,γxy,κx,κy,κz,κyz,κxz,κxy},$ where ε_x,ε_y,ε_z are membrane normal deformations, γ_yz,γ_zx,γ_xy are shear componets, and κ_x,κ_y,κ_z,κ_yz,κ_xz,κ_xy are curvatures. The generalized stress vector contains corresponding integral forces/moments: $s = {nx,ny,ny,vyz,vzx,vxy,mx,my, mz,myz,mxz, mxy}.$ Please note, for example, that bending moment m_x is defined as m_x = ∫ σ_xz dz, so it acts along the y-axis and positive value causes tension in bottom layer.
Nlgeo	0.
Status	Reliable

Table 17: tr_shell01 element summary

2.6.5 Quad1Mindlin Element

This class implements an quadrilateral, bilinear, four-node Mindlin plate. This type of element exhibit strong shear locking (thin plates exhibit almost no bending). Implements the lumped mass matrix. The element features are summarized in Table 18.


Keyword	quad1mindlin
Description	Quadrilateral, bilinear, four-node Mindlin plate
Specific parameters	[NIP #(in)]

Unknowns	Three dofs (w-displacement, u and v - rotation) are required in each node.
Approximation	Linear for all unknowns.
Integration	Default uses 4 integration points. No reduced integration is used, as it causes numerical problems.
Features	Layered cross section support.
CS properties	Cross section thickness is required.
Loads	Dead weight loads, and edge loads are supported.
Nlgeo	0.
Reference	[RobertCook1989]
Status	Experimental

Table 18: quad1mindlin element summary

2.6 Plate&Shell Elements

2.6.1 CCT Element

2.6.2 CCT3D Element

2.6.3 RerShell Element

2.6.4 tr_shell01 element

2.6.5 Quad1Mindlin Element